Analytic Number Theory: An Introduction by Richard Bellman

By Richard Bellman

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If the biggest term of r is r mXm and the biggest term of a is xn' then the degree of is less than the degree of r. So whence, setting q' = q + rmxm- n and r' desired modification of q and r. 5. Polynomials 31 Note that we split the conclusion into two cases: one where r is zero and one where it isn't. If we define the degree of 0 to be -1 or -oo, we can eliminate reference to whether r is zero or not, because then tl1e degree of 0 would be less than the degree of any nonzero polynomial. The division algorithm for polynomials can be performed in a standard long division format.

When is an integer m a unit in Zn, and when is an integers the inverse ofm in Zn? For example, 7·3 = 1 (mod 10) because 7·3 = 21 = 2·10+1, that is, 7 · 3- 1 is divisible by 10. In general, sm = 1 (mod n) exactly when sm - 1 is divisible by n. Writing this out explicitly, sm - 1 = tn for some integer t. Rearranging we get sm- tn = 1. Chapter 4. Units 50 But look at that equation it is essentially Bezout's equation for relatively prime integers m and n. From this equation we see that m and n are relatively prime, because any common divisor of m and n divides 1.

Thus from 111 we get 1-1 + 1 = 1, which is not a multiple of 11, so 111 is not a multiple of 11, but 1- 2 + 1 = 0, and 1- 1 + 1- 1 = 0, so 121 and 1111 are multiples of 11. Everyone knows that a number is a multiple of 5 exactly when its last digit is 0 or 5. Congruence explains why: modulo 5, every digit except the last digit represents 0. Exercises 1. = 0 such that ab = 1 or ab == 0. 2. Describe the integers that are congruent to 7 modulo 2. 3. Prove that if a= b (mod n) and b = c (mod n), then a= c (mod n).

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